I am looking the group of automorphisms $G$ of the curve definite over $\mathbb F_3$ by (in projective coordinates) $Y^2Z=X(X-Z)(X-2Z)$ Obviously, there are the automorphisms $X\mapsto X+\alpha Z$, $Y\mapsto\pm Y$ ($\alpha\in\mathbb F_3$) But are they the only ones? And a second question. What is the field $(\mathbb F_3(x)[y])^G$? I did not manage to determine it.

I am looking the group of automorphisms $G$ of the curve defined over $\mathbb F_3$ by (in projective coordinates) $Y^2Z=X(X-Z)(X-2Z)$. Obviously, there are the automorphisms $X\mapsto X+\alpha Z$ (for $\alpha\in\mathbb F_3$) and $Y\mapsto\pm Y$. But are they the only ones? And a second question: What is the field $(\mathbb F_3(x)[y])^G$? I did not manage to determine it.